A non empty subset H of a group G is a subgroup of G if, under the binary operation defined on G, H is a group.
In order to test whether a subset is a subgroup it is sufficient to complete the following three tests:
For example, the even integers are a subgroup of the integers under addition. Certainly the identity 0 is an even integer, the sum of two even integers is an even integer and the inverse of an even integer is an even integer.
The rotations and the identity in are a subgroup of . However, the reflections in together with the identity do not constitute a subgroup.
For any group G and , let be the subset defined by . This is a subgroup of G and is the cyclic subgroup generated by x. So, in the non zero real numbers under multiplication the set is a cyclic subgroup.
A very important result concerning subgroups of finite groups is Lagrange's Theorem which states that the order of a subgroup divides the order of a group.
With regard to finite cyclic groups, if G is a finite cyclic group of order n and x is a generator, then is a cyclic subgroup of order . Indeed, every subgroup of a cyclic group is cyclic. Now Lagrange's Theorem would tell us that a cyclic subgroup of a finite cyclic group of order n must have order a divisor of n. In this case, the converse is true, namely, for every divisor m of n there is a cyclic subgroup of order m. It is generated by where x is a generator of the cyclic group and k satisfies km = n. This converse statement is not true for groups in general.